MA.912.F.1.3

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.AR.2.2 
• MA.912.FL.3.4

Terms from the K-12 Glossary

• Rate of Change 
• Slope

Vertical Alignment

Previous Benchmarks

• MA.8.AR.3.2   
• MA.8.F.1.3

Next Benchmarks

• MA.912.F.1.4

Purpose and Instructional Strategies

• The purpose of this benchmark is to extend students’ understanding of rate of change to allow them to apply it in non-linear contexts. 
• Instruction emphasizes a graphical context so students can see the meaning of the average rate of change. Students can use graphing technology to help visualize this. 
  
  • Starting with the linear function $f$($x$) = 3$x$ − 2, shown below, ask students to calculate the rate of change between two points using the slope formula. Lead students to verify their calculations visually. 
  • Once students have successfully used the formula, transition to the graph of $f$($x$) = $x$². 
  • Highlight the same two points and ask students to discuss what the rate of change might be between them (MTR.4.1). Lead students realize that while there is not a constant rate of change, they can calculate an average rate of change for an interval. Show students that this is equivalent to calculating the slope of the line segment that connects the two points of interest. 

   

• Once students have an understanding, ask them to find the average rate of change for other intervals, such as −2 ≤ $x$ ≤ −1 or 0 ≤ $x$ ≤ 2. As each of these calculations produce different values, reinforce the concept that non-linear functions do not have constant rates of change (MTR.5.1). 
• Look for opportunities to continue students’ work with function notation. Ask students to find the average rate of change between $f$(1) and $f$(4) for $f$($x$).

Common Misconceptions or Errors

• Some graphs presented to students will only display a certain interval of data. Some students may mistakenly interpret the rate of change for that entire interval rather than a given sub-interval. In these cases, lead students to highlight the domain and range for the sub-interval requested so they can see it more clearly. 
• Students may be confused if the average rate of change is 0, even though the function is not constant. 
  • For example, the average rate of change of the function $f$($x$) =  $x$² from $x$ = −1 to $x$ = 1 is 0. 

Strategies to Support Tiered Instruction

• When determining an average rate of change on an interval where the function only increases or only decreases, instruction includes directions to highlight the domain and range for the interval so that the change can be seen more clearly. 
• Instruction includes providing the graph to visualize the change in the $x$- and $y$-values by drawing a line from the leftmost point on the graph within the interval to the rightmost point on the graph within the interval. Discuss with students how the average rate of change is the slope of the line between the two points. 
  • For example, for the function $f$($x$) = $x$² , on the interval [−2, 2], the average rate of change is zero. This can be visualized by drawing a line from the point (−2, 4) to (2, 4) which has slope of zero (horizontal line). 
• Instruction includes assistance recognizing the connection between slope (MA.912.AR.2) and rate of change for a linear function. A linear function has a constant rate of change. Regardless of the interval, the rate of change is the same (constant). For nonlinear functions rate of change is not constant, so it is not considered slope. However, the formula for slope can be used to calculate the average rate of change over an interval. 
• Teacher provides instruction on assigning the ordered pairs when calculating average rate of change. The formula for slope is the change in $y$ divided by the change in $x$. The designation of the interval points as the first or second pair of coordinates does not matter. 
• Teacher co-creates an $x$-$y$ chart to organize information when solving real-world problems involving a graphical representation. The $x$ column should be labeled with the input description used for the $x$-axis. The $y$ column should be labeled with the output description used for the $y$-axis.

Instructional Tasks

• Jorge invests an amount of $5,000 in a money market account at an annual interest rate of 5%, compounded monthly. The function $f$($x$) = 5000(1 + $\frac{\text{0.05}}{\text{12}}$)^12$x$, represents the value of the investment after $x$ years. 
  • Part A. Find the average rate of change in the value of Jorge’s investment between year 5 and year 7, between year 10 and year 12, and between year 15 and year 17. 
  • Part B. What do you notice about the change in value of the investment over each interval?

Instructional Items

• The graph below represents the number of producing gas wells in Kansas from 1989 to 2017. What is the average rate of change between 2000 and 2015?

Related Courses

Related Access Points

Related Resources

Student Resources

Parent Resources